The proposed research is in the field of tomography and has direct relevence for the uses of X-ray and other radiation in clinical diagnostic medicine. The goal of the research is to discover and implement an effective computer algorithm to recover the density of the part of the body outside of a specified region from tomographic data (line integrals) from X-ray beams that do not pass through that region. The same question will be studied for the plane integrals that are tomographic data gotten in one type of nuclear magnetic resonance zeugmatography. Each algorithm will be tested on mathematical phantoms and medical data. There are a number of compelling medical reasons for such an algorithm. For example, the beating heart creates error in regular chest tomograms, and bone or metal pins can create error in tomograms. It is important to be able to get good tomographic reconstructions of the organs around these regions. The way to do this is to develop the proposed algorithm. Two methods of solution are proposed, one of which is based on the singular value decomposition of Professor Quinto. Preliminary refinements are described that demonstrate the good potential of this method.